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Chapter 2: Q22P (page 52)

Find each of the following in rectangular $a + z b$ form if $z = 2 - z 3$ ; if $z = x + z y$ .
$\frac{1}{z - 1}$

Short Answer

Expert verified

The value for $\frac{1}{z - i}$ is given below

$\frac{1}{z - i} = \frac{1 + 2 i}{10} \frac{1}{z - i} = \frac{x - (y - 1) i}{x^{2} + {(y - 1)}^{2}}$

Step by step solution

01

Given Information

The complex number is $\frac{1}{z - i}$ .

02

Definition of the Complex number.

Complex number z can be written in the form of the real and imaginary numbers as:

$z = a + z b$

$z = a + z b$

Here z is the complex number, and a and b are real numbers

03

Find the value.

The complex number is $\frac{1}{z - 1}$ .

Find the values for 2-3i .

z=2-3i

$\frac{1}{z - i} = \frac{1}{2 - 4 i} = \frac{1}{2 - 4 i} 脳 \frac{2 + 4 i}{2 + 4 i} = \frac{2 + 4 i}{{(4 - 4 i)}^{2}} = \frac{1 + 2 i}{10}$

Find the values for x+iy .

$z = x + i y \frac{1}{z - 1} = \frac{1}{(y - 1) i} = \frac{1}{x + (y - 1)} 脳 \frac{x - (y - 1) i}{x - (y - 1) i} = \frac{x - (y - 1) i}{x^{2} + {(y - 1)}^{2}}$

The value for $\frac{1}{z - 1}$ is given below:

$\frac{1}{z - 1} = \frac{1 + 2 i}{10} \frac{1}{z - 1} = \frac{x - (y - 1) i}{x^{2} + {(y - 1)}^{2}}$

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Most popular questions from this chapter

Question:First simplify each of the following numbers to the x+iyform or to the $r e^{i 胃}$ form. Then plot the number in the complex plane.

$3 (c o s 28 掳 + i s i n 28 掳)$ .

Question:First simplify each of the following numbers to the x+iy form or to the $r e^{i 胃}$ form. Then plot the number in the complex plane.

1.6-2.7i .

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

$\frac{2 + 3 i}{1 - i}$ .

Describe geometrically the set of points in the complex plane satisfying the following equations.

$| z + 3 i | = 4$ .

Find each of the following in rectangular $a + {}_{z}b$ form $i f z = 2$ if ; if $z = x + {}_{z}y$ .

role="math" localid="1658486330935" $\frac{1}{z + 1}$ .

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